Problem: A point $(x,y)$ is randomly selected such that $0 \le x \le 3$ and $0 \le y \le 6$. What is the probability that $x+y \le 4$? Express your answer as a common fraction.
We draw the region and mark off the area where $x+y \le 4$:

[asy]
draw((0,0)--(3,0)--(3,6)--(0,6)--cycle);
fill((0,0)--(0,4)--(3,1)--(3,0)--cycle, gray(.7));
dot((0,0));
dot((3,0));
dot((0,6));
dot((3,6));
dot((0,4));
dot((3,1));
label("(0,0)", (0,0), W);
label("(0,6)", (0,6), W);
label("(0,4)", (0,4), W);
label("(3,1)", (3,1), E);
label("(3,0)", (3,0), E);
label("(3,6)", (3,6), E);
[/asy] The area of the rectangle is 18. The area of the shaded region, a trapezoid, is $\frac{1}{2}(1+4)\cdot3=\frac{15}{2}$. The probability that the point ends up in the shaded region is then $\boxed{\frac{5}{12}}$.